Could the Michelson-Morley experiment have detected gravitational waves? If everything went perfectly and they had no outside noise while conducting the experiment, could they have detected gravitational waves? What would it have looked like to them?
 A: Gravitational waves (GWs) cause space to stretch and shrink causing the distance between free falling objects to change as they pass.  In the Michelson and Morley experiment the mirrors were stuck in place on a pretty rigid sandstone slab.  Those mirrors are not freely falling.  So as the GWs warp space, the sandstone applies forces to the mirrors, resisting the change in distance.  In order for Michelson and Morley to detect gravitational waves their data analysis would have to account for the waves and the resistive forces from the sandstone.
So the Michelson and Morley experiment is a bad setup for detecting GWs.
But what if the mirrors were free to move about?  Could Michelson and Morley detected GWs with a small modification to their setup?
To see what Michelson and Morley could have detected, we make an order of magnitude estimate of the strain sensitivity of the Michelson--Morley interferometer (MMI).  We'll use some information from Wikipedia and fill in the gaps with reasonable guesses.
First an interferometer measures the gravitational wave strain,
$$h = \frac{\Delta L}{L},$$
where $L$ is the length of the path the light travels and $\Delta L$ is the change in length caused by the passing GW.  The MMI used multiple reflections to send the light a total of $L=11$ m.
The measurable $\Delta L$ is determined by the ability to measure the interference fringes.  Assuming no vibrational or thermal noise this is limited by photon shot noise.
$$\Delta L \sim \frac{\lambda}{\sqrt{N}},$$
where $\lambda$ is the wavelength of the light and $N$ is the number of photons collected during one GW cycle.
To get an intuitive feel why this is, lets think about why with one photon your resolution is about one wavelength. If you measure the photon at the end, you conclude there was constructive interference so the path length was the same for both paths.  If you don't measure the photon, there was destructive interference so one path was $\lambda/2$ longer than the other.  To accurately measure intermediate distances you need to see partial interference.  The more photons you start with, the better you can measure partial dimming caused by fractional path length changes.
The MMI used an oil lamp as a white light source.  Lets guess they focused about $100$ lumens into a $1$ cm$^2$ beam.  This is about $0.15$ W of light in the beam assuming $500$ nm light.
So how many photons can they collect during one cycle of the GW?
$$ N = \frac{P\, T}{E}, $$
Where $P$ is the power of the light, $T = 1/f_{GW}$ is the period of the GW, and $E=hc/\lambda$ is the energy of one photon of the light.
Lets assume the GW has a frequency of about $250$ Hz.  This was the frequency with the largest amplitude of GW150914.  The white light has a wavelength of about $500$ nm
$$ N = \frac{P\,\lambda}{hc\,f_{GW}} \sim 10^{15}$$
So the strain sensitivity is:
$$ h = \frac{\Delta L}{L} \sim \frac{\lambda}{\sqrt{N}\, L} \sim 10^{-15} $$
GW150914 had a peak strain amplitude of about $10^{-21}$ at 250 Hz.  So the MMI is six orders of magnitude short.
If we had used a $10$ W laser, and increased the effective light travel distance to $1000$ km (many bounces back and forth in km scale arms) we'd have LIGO.  And then we could detect it.
But there was one big caveat!  We assumed no vibrational noise!  The MMI was in a basement,  built on a single sandstone slab, floating in a pool of mercury.  This is very cool.  It also damped vibrational noise sufficiently for their experiment.  But it is nowhere near the vibrational damping necessary to actually achieve the nominal $h\sim 10^{-15}$ sensitivity we estimated in practice.
A: In practice if the conditions had been perfect - no, absolutely not. LIGO has the precision of being able to measure a distance change of +/- width of a hair on a scale of a few lightyears. Even if a gravitational wave is larger by a few orders of magnitude the Michelson-Morley experiment was nowhere in the same universe of precision.
Theoretically, if the wavelength of the gravitational wave is as big or bigger than the distance between the mirror/splitters and they had the precision necessary (or wave big enough), then they could have detected one. It uses a very similar principle as LIGO - in fact LIGO compares their interferometer to the Michelson-Morley interferometer in their first paper.
A: If the gravitational wave is strong enough you should be able to feel it without a detector. Detection is a question of signal to noise ratio.
